How to find the fourth term of (y- 3x)^6?

To find the fourth term of \((y - 3x)^6\), we can use the *Binomial Theorem*. The theorem states that:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, \(a = y\), \(b = -3x\), and \(n = 6\). The general term (or the \((k+1)\)-th term) of the expansion is given by:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

### Step 1: Identify the fourth term

For the fourth term, we need \(k = 3\) (since the first term corresponds to \(k = 0\)). Thus, the fourth term is given by:

\[
T_4 = \binom{6}{3} y^{6-3} (-3x)^3
\]

### Step 2: Calculate each part

1. **Calculate the binomial coefficient**:

   \[
   \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
   \]

2. **Calculate \(y^{6-3}\)**:

   \[
   y^{6-3} = y^3
   \]

3. **Calculate \((-3x)^3\)**:

   \[
   (-3x)^3 = -27x^3
   \]

### Step 3: Multiply everything together

\[
T_4 = 20 \cdot y^3 \cdot (-27x^3) = -540 y^3 x^3
\]

### Final Answer
The fourth term of \((y - 3x)^6\) is:

\[
\boxed{-540 y^3 x^3}
\]

Would you like further details on any step, or have any other questions?

---

1. What is the general term formula for a binomial expansion?
2. How to find the coefficient of a specific term in a binomial expression?
3. How does changing the sign inside the binomial affect the terms?
4. What would happen if we expand \((3x - y)^6\) instead?
5. How does the exponent impact the complexity of the expansion?

**Tip:** The power and coefficients in binomial expansions can be quickly identified by understanding patterns in Pascal's triangle.

How to find the fourth term of (y - 3x)^6?

Learn how to find the fourth term of (y - 3x)^6 using the Binomial Theorem. This step-by-step solution covers essential algebraic concepts, including binomial expansion and the general term formula, making it suitable for Grades 10-12.

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